33_580ln_fa08

33_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.80 Lecture #3 3 Fall, 2008 Page 1 of 10 pages Lecture #3 3 : Vibronic Coupling Last time: H 2 CO A 1 A 2 X 1 A 1 Electronically forbidden if A -state is planar vibronically allowed to alternate v 4 vibrational levels if A -state is planar inertial defect says A -state is not planar expect to see all v 4 if not planar staggering of v 4 level spacings inversion through low barrier to planarity dynamic vs. rigid molecule symmetry classification: molecular symmetry group How does vibronic coupling really work? What are the vibrational intensity factors analogous to Franck-Condon factors in the case of vibronically allowed rather than electronically allowed transition? See T. Azumi and K. Matsuzaki, Photochemistry and Photobiology 25 , 315 (1977) for an extremely readable review article. Outline: Crude Adiabatic Approximation Correction of ψ for effect of neglected off-diagonal matrix elements H 2 CO A 1 A 2 example What happens to Franck-Condon factors for a “vibronically allowed” transition? Two electronic basis sets — prediagonalize “symmetry-breaking” vibronic interaction Changes in shapes of potential curves (deperturb to a simpler, “more natural” shape) K. K. Innes’ model for vibrational band intensities and level staggering Recall Born-Oppenheimer or “clamped nuclei” approximation. We use this procedure to define complete sets of electronic and nuclear motion wavefunctions with which we can FORMALLY expand exact ψ ’s and compute (or parametrize) all properties of exact eigenstates. The simplest basis set is called “CRUDE ADIABATIC” (CA) CA o CA ψ jt ( r, Q ) = ψ j ( r, Q 0 ) χ jt ( ) Q vibrational state fixed nuclear locations! electronic state Q 0 is a convenient reference structure (usually the equilibrium geometry or a high-symmetry potential energy maximum or saddle point). o ψ j is the electronic wavefunction in the j-th electronic state computed at the chosen and explicitly specified set of fixed nuclear coordinates Q 0 .
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5.80 Lecture #3 3 Fall, 2008 Page 2 of 10 pages CA ( Q ) is the vibration-rotation wavefunction computed from an approximate nuclear Schrödinger χ jt Equation. eigenvalue of clamped nuclei nuclear potential electronic U(r, Q ) = U(r, Q ) – U(r, Q 0 ) kinetic energy of Schrödinger change in e nuclear and e e energy bare nuclei Equation at Q 0 Coulomb energy o o o CA CA T N ( Q ) + V( Q ) + ε j ( Q 0 ) + ψ j ( r, Q 0 ) U(r, Q ) ψ j ( r, Q 0 ) χ jt ( ) Q = E jt Q CA χ jt ( ) effective potential- energy surface Note that the U integral is evaluated using ψ o j (r, Q 0 ) thus cannot contain the exact effect of distortion of molecule from Q 0 . To get a better representation of the distortion from Q 0 , we must use perturbation theory.
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33_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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